On the Mechanization of the Proof of Hessenberg’s Theorem in Coherent Logic

2016

J Autom Reasoning

In this paper, we report on the formalization of a synthetic proof of Pappus' theorem. We provide two versions of the theorem: the first one is proved in neutral geometry (without assuming the parallel postulate), the second (usual) version is proved in Euclidean geometry. The proof that we formalize is the one presented by Hilbert in The Foundations of Geometry, which has been described in detail by Schwabhäuser, Szmielew and Tarski in part I of Metamathematische Methoden in der Geometrie. We highlight the steps that are still missing in this later version. The proofs are checked formally using the Coq proof assistant. Our proofs are based on Tarski's axiom system for geometry without any continuity axiom. This theorem is an important milestone toward obtaining the arithmetization of geometry, which will allow us to provide a connection between analytic and synthetic geometry.

Section: Related Work: Other Formal Proofs Related To Pappus' Theoremmentioning

confidence: 99%

A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry

Braun

2016

J Autom Reasoning

“…Certain fragments of first-order logic allow automated theorem proving with a simple production of readable proofs -one of them is coherent logic. Coherent logic (CL), initially defined by Skolem, has gained new attention in recent years [5,12,6]. It consists of formulae of the following form:…”

Section: Contextmentioning

confidence: 99%

“…The formalization by Beeson and Wos can be considered semi-automatic as the authors provide many guidelines to Otter for the difficult theorems: they provide the points to be constructed and also long lists of proof steps. Beeson and Wos also provide intermediate lemmas not appearing in the original book, some of which seem to be inspired by the Coq formalization (see the lemma in Chapter 5 6 ). Within this development, neither formal proofs (verifiable by proof assistants) nor readable and natural-language proofs were considered.…”

Section: Overview Of the Existing Computer Developmentsmentioning

confidence: 99%

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

Đurđević

Janicic

2015

Ann Math Artif Intell

The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of nontrivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics-a book on Tarski's geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37% of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42%. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge.

“…lines) are distinct. As suggested in [2], axioms 'Point Uniqueness' and 'Line Uniqueness' can be merged into a more convenient axiom with no negation. This axiom is classically equivalent to the conjunction of the two others:…”

Section: A First Set Of Axiomsmentioning

confidence: 99%

“…Table 1. The incidence relation of PG (2,5). Each column lists the lines incident to the given point.…”

Section: E Lines As Set Of Pointsmentioning

confidence: 99%

Formalizing Projective Plane Geometry in Coq

Magaud

Automated Deduction in Geometry

Schreck

2011

We investigate how projective plane geometry can be formalized in a proof assistant such as Coq. Such a formalization increases the reliability of textbook proofs whose details and particular cases are often overlooked and left to the reader as exercises. Projective plane geometry is described through two different axiom systems which are formally proved equivalent. Usual properties such as decidability of equality of points (and lines) are then proved in a constructive way. The duality principle as well as formal models of projective plane geometry are then studied and implemented in Coq. Finally, we formally prove in Coq that Desargues' property is independent of the axioms of projective plane geometry.